Theorem dilsetN 39658

Description: The set of dilations for a fiducial atom 𝐷. (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
dilset.a	⊢ 𝐴 = (Atoms‘𝐾)
dilset.s	⊢ 𝑆 = (PSubSp‘𝐾)
dilset.w	⊢ 𝑊 = (WAtoms‘𝐾)
dilset.m	⊢ 𝑀 = (PAut‘𝐾)
dilset.l	⊢ 𝐿 = (Dil‘𝐾)

Assertion

Ref	Expression
dilsetN	⊢ ((𝐾 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴) → (𝐿‘𝐷) = {𝑓 ∈ 𝑀 ∣ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝐷) → (𝑓‘𝑥) = 𝑥)})

Distinct variable groups:   𝑥,𝑓,𝐾   𝑓,𝑀   𝑥,𝑆   𝐷,𝑓,𝑥

Allowed substitution hints:   𝐴(𝑥,𝑓)   𝐵(𝑥,𝑓)   𝑆(𝑓)   𝐿(𝑥,𝑓)   𝑀(𝑥)   𝑊(𝑥,𝑓)

Proof of Theorem dilsetN

Dummy variable 𝑑 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dilset.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
2		dilset.s	. . . 4 ⊢ 𝑆 = (PSubSp‘𝐾)
3		dilset.w	. . . 4 ⊢ 𝑊 = (WAtoms‘𝐾)
4		dilset.m	. . . 4 ⊢ 𝑀 = (PAut‘𝐾)
5		dilset.l	. . . 4 ⊢ 𝐿 = (Dil‘𝐾)
6	1, 2, 3, 4, 5	dilfsetN 39657	. . 3 ⊢ (𝐾 ∈ 𝐵 → 𝐿 = (𝑑 ∈ 𝐴 ↦ {𝑓 ∈ 𝑀 ∣ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝑑) → (𝑓‘𝑥) = 𝑥)}))
7	6	fveq1d 6904	. 2 ⊢ (𝐾 ∈ 𝐵 → (𝐿‘𝐷) = ((𝑑 ∈ 𝐴 ↦ {𝑓 ∈ 𝑀 ∣ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝑑) → (𝑓‘𝑥) = 𝑥)})‘𝐷))
8		fveq2 6902	. . . . . . 7 ⊢ (𝑑 = 𝐷 → (𝑊‘𝑑) = (𝑊‘𝐷))
9	8	sseq2d 4014	. . . . . 6 ⊢ (𝑑 = 𝐷 → (𝑥 ⊆ (𝑊‘𝑑) ↔ 𝑥 ⊆ (𝑊‘𝐷)))
10	9	imbi1d 340	. . . . 5 ⊢ (𝑑 = 𝐷 → ((𝑥 ⊆ (𝑊‘𝑑) → (𝑓‘𝑥) = 𝑥) ↔ (𝑥 ⊆ (𝑊‘𝐷) → (𝑓‘𝑥) = 𝑥)))
11	10	ralbidv 3175	. . . 4 ⊢ (𝑑 = 𝐷 → (∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝑑) → (𝑓‘𝑥) = 𝑥) ↔ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝐷) → (𝑓‘𝑥) = 𝑥)))
12	11	rabbidv 3438	. . 3 ⊢ (𝑑 = 𝐷 → {𝑓 ∈ 𝑀 ∣ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝑑) → (𝑓‘𝑥) = 𝑥)} = {𝑓 ∈ 𝑀 ∣ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝐷) → (𝑓‘𝑥) = 𝑥)})
13		eqid 2728	. . 3 ⊢ (𝑑 ∈ 𝐴 ↦ {𝑓 ∈ 𝑀 ∣ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝑑) → (𝑓‘𝑥) = 𝑥)}) = (𝑑 ∈ 𝐴 ↦ {𝑓 ∈ 𝑀 ∣ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝑑) → (𝑓‘𝑥) = 𝑥)})
14	4	fvexi 6916	. . . 4 ⊢ 𝑀 ∈ V
15	14	rabex 5338	. . 3 ⊢ {𝑓 ∈ 𝑀 ∣ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝐷) → (𝑓‘𝑥) = 𝑥)} ∈ V
16	12, 13, 15	fvmpt 7010	. 2 ⊢ (𝐷 ∈ 𝐴 → ((𝑑 ∈ 𝐴 ↦ {𝑓 ∈ 𝑀 ∣ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝑑) → (𝑓‘𝑥) = 𝑥)})‘𝐷) = {𝑓 ∈ 𝑀 ∣ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝐷) → (𝑓‘𝑥) = 𝑥)})
17	7, 16	sylan9eq 2788	1 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴) → (𝐿‘𝐷) = {𝑓 ∈ 𝑀 ∣ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝐷) → (𝑓‘𝑥) = 𝑥)})

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∀wral 3058  {crab 3430   ⊆ wss 3949   ↦ cmpt 5235  ‘cfv 6553  Atomscatm 38767  PSubSpcpsubsp 39001  WAtomscwpointsN 39491  PAutcpautN 39492  DilcdilN 39607

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-dilN 39611

This theorem is referenced by:  isdilN  39659